Phase Transitions in Polymers and Their Analysis

Phase Transitions in Polymers and Their Analysis

Michael Hess Department of Physical Chemistry

4th UNESCO SCHOOL on MACROMOLECULES AND MATERIALS, UNIVERSITY of STELLENBOSCH 07.04.-08.04.2001 Talking about Phase Diagrams: a) p-V-T Diagrams of pure components b) p(x)-, T(x)-, Bakhuise-Roozeboom-Diagrams of binary (multicomponent) mixtures

miscible systems partially miscible systems systems with an eutecticum systems with UCP and/or LCP Binary Phase Diagram with a Liquid Crystal

100% amorphous 100% crystalline semi-crystalline The Gibbs energy energy at a phase transition

What makes things go:

∆transG < 0

Calorimeter response

All is free volume and chain mobility

V liquid

glass free volume

occupied volume

Tg T The glassy state can be described as an iso-free volume state The GNOMIX device for p-v-T experiments

Example for a Strong Pressure/Volume Effect of a LC-Transition

Temperature

[Pa-1] [Pa-1] Isothermal Compressibility of PET p-v-T measurements are useful for:

• Basing on the free-volume concepts it is possible to predict service performance

and service life of polymeric materials.

• Polymer/polymer miscibility can be predicted.

• Chemical reactions can be followed provided they are accompanied by volume-

effects

• Processing parameters can be optimized without a trial and error procedure

• The surface tension of polymer melts can be estimated. In all cases one works with two key quantities:

the isobaric expansivity

1  ∂ V  α=   ∂ V T P

the isothermal compressibility

1  ∂ V  κ =−   ∂ V P T

in many cases Ehrenfest's equation holds and the pressure dependence of the glass transition temperature is given by:

d Tg = Δκ dP Δα

although the glass transition is not a second order transition. Differential Equations at Phase Transitions

According to Ehrenfest, thermodynamic transitions are classified according to discontinuities in the derivatives of the Gibbs Energy

 ∂G   ∂G  G = H - T⋅⋅⋅S; dG =Vdp−SdT=  dp+  dT ∂  ∂   p T T p

 ∂G   ∂2G   ∂S  1  ∂Q    =−S;   = −  =−   ∂   ∂  ∂   T  p  ∂T2   T p T T p p $!# !" cp cp ≡ heat capacity at constant pressure

 ∂G    =V  ∂   p  T  ∂2G   ∂V    =   = α*  ∂p⋅∂T   ∂T    p p

α*≡ isobaric expansivity, correspondingly β* ≡ isothermal compressibility

1. Order Transition (Discontinuous Thermodynamic Transition)

Bend in G(T) jump in S(T), V(T) and H(T)

2. Order Transition (Continuous Thermodynamic Transition)

Bend in V(T), H(T) and S(T) jump in α*, κ* and cp

The Glass Transition (Continuous “Freezing in”, a Kinetic Effect)

Bend in V(T), H(T) and S(T) jump in α*, κ* and cp Shape different from a 2nd order transition First order thermodynamic transitions follow the Clausius-Clapeyron Equation:

dp = ΔΗ⋅ 1 dt ΔV T

For second order thermodynamic transitions Ehrenfest`s Equations are valid

  Δα  dP  = tr    dT  tr Δκtr

  (c )  dP  = P tr    dT  ⋅ ⋅ tr Ttr (Vspez.)tr Δαtr tr Δα ⋅ tr tr ) tr tr ) P Δα Δκ (c spez. = (V tr ⋅      tr T dP dT

g      = tr      dP dT     

DP 1.3 at T at 1.3 DP and

DP 1.2 DP , ) ) ) ∆κ / Glas Glas ∆κ Glas ∆κ / / ∆α ( ∆α ∆α ( (

DP 1.1 DP (d P /d T ) (d P /d T )

(d P /d T ) f(P ) D P1 .2 f(P ) D P1 .3 f(P ) D P1 .1 pressure P [bar] pressure P [bar] pressure P [bar] pressure ) ) ) ∆α ∆α ∆α spez. spez. spez. / T V / T V / T V P P P c c c ∆ ∆ ∆ ( ( ( 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 200 400 600 800 1000 1200 1400 1600 1800 2000 8

36 34 32 30 28 26 24 22 20 18 16 14 12 10 34 32 30 28 26 24 22 20 18 16 14 12 10 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15

( barK] r/K a [b ) f(P ] r/K a [b ) f(P ] r/K a [b ) f(P

Ehrenfest's RelationsEhrenfest's of Bakhuise-Roozeboom-Diagram x-T projections

Bakhuise-Roozeboom-Diagram Thermo-Mechanical Analysis

Expansivity: the free volume increases with temperature

Expansion [mm]

Tg T Penetration:

Penetration [mm]

Tg T Dynamic-Mechanical Analysis (Free Oscillating Torsion)

Periodic torsional stress τ periodic deformation γ phase shift δ τ = iωt γ =γ i()ωt−δ τ0 e 0 e

complex torsional modulus τ τ τ τ * = = 0 iδ = 0 cos δ + i⋅ 0 ⋅sin δ G γ γ e γ γ 0 $!#0 !" $!#0 !" G' G''

Differential equation of the oscillating system: f⋅G'' f⋅G'+f ϕ..()t +a⋅ϕ. t +b⋅ϕ ()t = 0; a = b = 1 Θ⋅ Θ ωe f ≡ form factor of the sample; f1 ≡ form factor of the spring ωe ≡ frequency of the oscillating system Θ ≡ momentum of inertia γ0 σ0

ϑω-1 2 πω-1 stress

time φ 4 0

φ 1 t, s   2 φ  ϕ  2 ()t Λ≡ln   ϕ π  2 (t)  +  ´ 0

t Φ Φ    Φ Φ   ωe 

-2

1/ν 1/ν -4 the free damped oscillation is then described by:

−ω ⋅Λ ϕ()= ϕ ⋅ e ⋅t Λ t 0 e 2π with the logarithmic decrement of the     ϕ()t damped oscillation Λ ≡ln  , so that finally:   2π  ϕt+    ωe 

Θ   Θ Λ   G' = ω2−ω2 and G'' = ⋅ω2⋅ f  e 0 f e π

ω0 is the frequency of the free oscillating system without sample ωe is the frequency of the oscillating system with sample

The loss tangent or damping is finally defined by: G'' tan δ ≡ G'

glass-transition, melting and glass-rubber transition

General Aspects of Binary Phase Diagrams

1. 2 two-phase regions must be separated by either a bivariant one-phase range, or by a part of a non-variant three-phase line.

2. 2 one-phase regions cannot be adjacted, they must be separated by a two-phase region.

3. If there is 2 two-phase region on one side of a three-phase line, there must be 2 two-phase areas on the other side of that line.

4. Metastable extensions beyond the three- phase line must fall within the two-phase ranges in the areas they extend into. Binary Isobaric Phase Diagram

n o n -v a rian t th re e p hase eq u ilib riu m Binary Isobaric Phase Diagram

n o n -v a rian t th re e p hase eq u ilib riu m B inary Isobaric Phase D iagram w ith U C ST and L C ST liq uidus line solidus line

solvus line

eutecticum

Water PEO

concentration dep ending T g

I liquid phase, hom ogeneous solution II two-phase area, ice/hom ogeneous solution III two-phase area, mixed crystal/hom ogeneous solution IV one-phase area, m ixed crystal, PEO -rich V two-phase area, ice/m ixed crystal Φ T g concentration independent glass-transition tem perature of the over-saturated solution Ψ T g concentration independent glass-transition tem perature of eutectic com position Acknow ledgem ent

D r. P a u l Z o lle r, B o u ld e r, C o lo ra d o , is g re a tly a c k n o w le d g e d fo r p ro v id in g th e s k e tc h o f th e GNOMIX m achine and the perm ission to reproduce it here. D r. Pionteck and his cow orker M r. Kunath, In stitu te fo r P o lym e r R e se a rch , D re sd e n , G e rm a n y, perform ed the p-v-T-m easurem ents and provided helpful discussions. Prof. Borchard, Departm ent of Applied Physical Chem istry, Gerhard-M ercator-University D uisburg, and his cow orker Dobnik provided the phase d iag ra m o f P E O /w a te r. D r. W o e lk e , M r. M e y e r, a n d M rs. S c h w a rk , Departm ent of Physical Chem istry, Gerhard- M e rca to r-U n ive rsity D u isb u rg , G e rm a n y, w e re in v o lv e d in s y n th e th ic w o rk , D M T A - a n d D S C - m easurem ents, and prepared m any of the figures. Finally thanks are due to C. P. Lee, Kw ang-Ju, South Korea, for helpful discussions.